Sparse Gaussian Random Symmetrical Ensemble has been introduced. Its statistical properties have been investigated. Transition from chaotic sub-ensemble to integrable one versus fraction of the matrix elements not equal to zero has been found. Jump of cumulative distribution function at energy \(E=0\) has been interpreted as an order parameter which differentiates between chaotic and integrable ensembles.

For a simplicial manifold we construct the differential geometry structure and use it to investigate linear connections, metric and gravity. We discuss and compare three main approaches and calculate the resulting gravity action functionals.

In this paper a new approach to solving the Ising–Onsager problem in external magnetic field is investigated. The expression for free energy per Ising spin in external field both for the two dimensional and three dimensional Ising model with interaction of the nearest neighbors are derived. The representations of free energy being expressed by multidimensional integrals of Gauss type with the appropriate dimensionality are shown. The possibility of calculating the integrals and the critical indices on the base of the derived representations for free energy is investigated.

The class of two-particle relativistic models which are described by the time-asymmetric Fokker-type integral of general form is considered. The manifestly covariant description of such models is constructed in the framework of the canonical formalism with constraints. By means of certain gauge fixing, the time-asymmetric models are reduced to the Bakamjian– Thomas model supplemented by a space-time interpretation. The corresponding two-body problem is reduced to quadratures.

Multifractal scaling analysis of nuclear giant resonance transition probability distributions is performed within the approximation which takes into account the one-particle–one-hole (l\(p\)–l\(h\)) and 2\(p\)–2\(h\) states. A new measure to determine the fractal dimensions of the spectra is introduced. It is found that chaotic dynamics governing the decay leads to non-trivial multifractal scaling properties. Such a kind of scaling is absent in the case of regular dynamics. The degree of collectivity is another element which worsens the scaling.

Fourdimensional bicovariant differential *-calculus on quantum E(2) group is constructed. The relevant Lie algebra is obtained and covariant differential calculus on quantum plane is found.

A recently proposed integral representation for permanents is rederived using only elementary combinatorics. For this proof the assumption that the matrix, for which the permanent is calculated, has an inverse is not necessary.

The real version of the wave equations that control the propagation of Maxwell–Dirac fields in complex Minkowski space is presented. It is particularly shown that the electromagnetic potential turns out to be subject to certain pluriharmonicity conditions whenever the Maxwell fields are taken to carry positive energy. The actual derivation of a set of invariant exactness relations for the entire system is then carried out.

We show that the kinematic reconstruction of the \(e^+e^- \to W^+W^- \to jj \tau \bar \nu _{\tau }\) events have a one-parameter ambiguity when reconstructed from the momentum of all measured \(W^-\) decay products. We propose a hybrid method of reconstruction of the \(e^+e^- \to W^+W^- \to jj \tau \bar \nu _{\tau }\) events. This is based on the observation that the difference between the \(\tau \) production angles and the production angles of the sum of its visible decay products is small, whilst the \(\tau \) energy is poorly reconstructed. This method consists of taking the \(\tau \) production angles from those measured for the sum of the visible \(\tau \) decay products and reconstructing the \(\tau \) energy from energy-momentum conservation constraints. A reconstruction using this method is found to be well-defined and possess a unique solution for the \(\tau \) momentum range at LEP2 and NLC.

In this thesis exact results on \(\mathcal O(\alpha ^2)\) single bremsstrahlung corrections to low angle Bhabha scattering at LEP/SLC energies are given. The calculation represents the last outstanding theoretical second order subleading electroweak contribution for that process, needed to determine the experimental luminosity at second generation LEP detectors below the 0.1% precision threshold. The exact, fully differential result is obtained by employing analytical as well as computer-algebraic methods and includes terms up to \(\mathcal O\)(0.05%) relative to the Born cross section. The initial output of over 20,000 terms could be reduced to 90, only 18 of which are shown to be numerically relevant and for which a simple logarithmic ansatz is derived, that is in remarkable agreement with the complete answer. Strong consistency checks are performed, including Ward–Takahashi identities and tests on the right infrared limit according to the Yennie, Frautschi and Suura program. Monte Carlo results for the integrated cross section are compared with existing calculations in the leading logarithmic approximation for a chosen set of experimental cuts. The size of the missing subleading terms is found to be small but, non negligible in the context of setting stringent limits on Standard Model predictions and thus its realm of validity.

We apply the bunching-parameter analysis to the hadron–hadron collisions within the FRITIOF model. The monofractal structure of intermittency is observed, in contrast to the multifractal structure in the \(e^+e^-\) annihilation. The unusual enhancement of the second-order bunching parameter is a direct manifestation of the enhanced void probability.

A two-body wave equation is derived, corresponding to the hypothesis (discussed already in the past) that \(u\) and \(d\) current quarks are relativistic bound states of a spin-1/2 preon existing in two weak flavors and three colors, and a spin-0 preon with no weak flavor nor color, held together by a new strong but Abelian, vectorlike gauge force. Some nonconventional (though somewhat nostalgic) consequences of this strong Abelian binding within composite quarks are pointed out. Among them are: new tiny magnetic-type moments of quarks (and nucleons) and new isomeric nucleon states possibly excitable at some high energies. The latter may arise through a rearrangement mechanism for quark preons inside nucleons. In the interaction \(q \bar q \to q \bar q\) of preon-composite quarks, beside the color forces, there act additional exchange forces corresponding to diagrams analogical to the so called dual diagrams for the interaction \(\pi \pi \to \pi \pi \) of quark-composite pions.

The contribution of highly asymmetric \(q- \bar q\) configurations to the onium–onium scattering at high energy is discussed in the framework of Mueller’s QCD dipole picture. A modification of Mueller’s formula is proposed and applied to deep inelastic lepton–onium scattering.

The structure of low-lying states of nuclei with four active nucleons in the 1\(s\)0d and 1\(p\)0f shells is studied in the framework of the Collective Pair Approximation. The collective pairs determined by diagonalizing the Hamiltonian in the space of two nucleons outside closed shells are considered as building blocks to describe a nucleus with 2\(n\) valence nucleons in terms of \(n\) pairs. It is shown that the low-lying spectrum can be described quite well by considering only a selected subset of all possible collective pairs.